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Find Asymptotes, Critical and Inflection Points

This example describes how to analyze a simple function to find its asymptotes, maximum, minimum, and inflection point.

Define a Function

The function in this example is

f(x)=3x2+6x1x2+x3.

To create the function, enter the following commands:

syms x
num = 3*x^2 + 6*x -1;
denom = x^2 + x - 3;
f = num/denom
f = 
(3*x^2 + 6*x - 1)/(x^2 + x - 3)

Plot the function f by using fplot. The fplot function automatically shows horizontal asymptotes.

fplot(f)

Find Asymptotes

To mathematically find the horizontal asymptote of f, take the limit of f as x approaches positive infinity:

limit(f, inf)
ans = 
3

The limit as x approaches negative infinity is also 3. This result means the line y = 3 is a horizontal asymptote to f.

To find the vertical asymptotes of f, set the denominator equal to 0 and solve by entering the following command:

roots = solve(denom)
roots =
 - 13^(1/2)/2 - 1/2
   13^(1/2)/2 - 1/2

Note

MATLAB® does not always return the roots to an equation in the same order.

roots indicates that the vertical asymptotes are the lines

x=1+132,

and

x=1132.

Find Maximum and Minimum

You can see from the graph that f has a local maximum between the points x = –2 and x = 0, and might have a local minimum between x = –6 and x = –2. To find the x-coordinates of the maximum and minimum, first take the derivative of f:

f1 = diff(f)
f1 = 
(6*x + 6)/(x^2 + x - 3) - ((2*x + 1)*(3*x^2 + 6*x - 1))/(x^2 + x - 3)^2

To simplify this expression, enter

f1 = simplify(f1)
f1 =
 -(3*x^2 + 16*x + 17)/(x^2 + x - 3)^2

Next, set the derivative equal to 0 and solve for the critical points:

crit_pts = solve(f1)
crit_pts =
 - 13^(1/2)/3 - 8/3
   13^(1/2)/3 - 8/3

It is clear from the graph of f that it has a local minimum at

x1=8133,

and a local maximum at

x2=8+133.

You can plot the maximum and minimum of f with the following commands:

fplot(f)
hold on
plot(double(crit_pts), double(subs(f,crit_pts)),'ro')
title('Maximum and Minimum of f')
text(-4.8,5.5,'Local minimum')
text(-2,4,'Local maximum')
hold off

Find Inflection Point

To find the inflection point of f, set the second derivative equal to 0 and solve.

f2 = diff(f1);
inflec_pt = solve(f2,'MaxDegree',3);
double(inflec_pt)

This returns

ans =
  -5.2635 + 0.0000i
  -1.3682 - 0.8511i
  -1.3682 + 0.8511i

In this example, only the first element is a real number, so this is the only inflection point. The order of the roots can vary.

Rather than selecting the real root by indexing into inter_pt, identify the real root by determining which roots have a zero-valued imaginary part.

idx = imag(double(inflec_pt)) == 0;
inflec_pt = inflec_pt(idx);

To obtain the value of the inflection point, enter

vpa(inflec_pt)
ans =
 
-5.2635217342053210183437823783747

Plot the inflection point. The extra argument, [-9 6], in fplot extends the range of x values in the plot so that you see the inflection point more clearly, as shown in the following figure.

fplot(f, [-9 6])
hold on
plot(double(inflec_pt), double(subs(f,inflec_pt)),'ro')
title('Inflection Point of f')
text(-7,1,'Inflection point')
hold off